FRONTIERS OF
EVOLUTIONARY COMPUTATION
Genetic Algorithms and Evolutionary Computation
Consulting Editor, David E. Goldberg
University of Illinois at Urbana-Champaign [email protected]
Additional titles in the series:
Efficient and Accurate Parallel Genetic Algorithms, Erick Cantú-Paz ISBN: 0- 7923-7221-2
Estimation of Distribution Algorithms: A New Tool for Evolutionary
Computation, edited by Pedro Larrañaga, Jose A. Lozano ISBN: 0-7923-7466-5 Evolutionary Optimization in Dynamic Environments, Jürgen Branke ISBN: 0- 7923-7631-5
Anticipatory Learning Classifier Systems, Martin V. Butz ISBN: 0-7923-7630-7 Evolutionary Algorithms for Solving Multi-Objective Problems, Carlos A. Coello Coello, David A. Van Veldhuizen, and Gary B. Lamont ISBN: 0-306-46762-3 OmeGA: A Competent Genetic Algorithm for Solving Permutation and Scheduling Problems, Dimitri Knjazew ISBN: 0-7923-7460-6
The Design of Innovation: Lessons from and for Competent Genetic Algorithms, David E. Goldberg ISBN: 1-4020-7098-5
Noisy Optimization with Evolution Strategies, Dirk V. Arnold ISBN: 1 -4020- 7105-1
Classical and Evolutionary Algorithms in the Optimization of Optical Systems, Darko ISBN: 1-4020- 7140-X
Evolutionary Algorithms for Embedded System Design, edited by Rolf Drechsler, Nicole Drechsler: ISBN: 1-4020- 7276-7
Genetic Algorithms and Evolutionary Computation publishes research monographs, edited collections, and graduate-level texts in this rapidly growing field. Primary areas of coverage include the theory, implementation, and application of genetic algorithms (GAs), evolution strategies (ESs), evolutionary programming (EP), learning classifier systems (LCSs) and other variants of genetic and evolutionary computation (GEC). Proposals in related fields
such as artificial life, adaptive behavior, artificial immune
systems, agent-based systems, neural computing, fuzzy GENAGENAGENA
systems, and quantum computing will be considered for GENAGENAGENA
publication in this series as long as GEC techniques are part of Genetic Algorithms and
or inspiration for the system being described. Manuscripts Evolutionary Computation
describing GEC applications in all areas of engineering,
commerce, the sciences, and the humanities are encouraged. http://www.wkap.nl/prod/s/GENA
FRONTIERS OF
EVOLUTIONARY COMPUTATION
edited by
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List of Figures xi
List of Tables xiii
Preface xv
Contributing Authors xvii
1
Towards a Theory of Organisms and Evolving Automata 1 Heinz Mühlenbein
1 Introduction 1
2 Evolutionary computation and theories of evolution 3
3 Darwin’s continental cycle conjecture 5
4 The system view of evolution 7
5 Von Neumann’s self-reproducing automata 9
6 Turing’s intelligent machine 11
7 What can be computed by an artificial neural network? 13
8 Limits of computing and common sense 14
9 A logical theory of adaptive systems 16
10 The for creating artificial intelligence 19
11 Probabilistic logic 20
11.1 Von Neumann’s probabilistic logics 20 11.2 The conditional probability computer 21
11.3 Modern probabilistic logic 22
12 Stochastic analysis of cellular automata 24
12.1 The nonlinear voter model 24
12.2 Stochastic analysis of one dimensional SCA 26 13 Stochastic analysis of evolutionary algorithms 27
13.1 Boltzmann selection 29
13.2 Factorization of the distribution 29 13.3 Holland’s schema analysis and the Boltzmann distribu
tion 31
14 Stochastic analysis and symbolic representations 33
15 Conclusion 33
2
Two Grand Challenges for EC 37
Kenneth De Jong
1 Introduction 37
2 Historical Diversity 38
3 The Challenge of Unification 39
3.1 Modeling the Dynamics of Population Evolution 40
3.1.1 Choosing Population Sizes 40
3.1.2 Deletion Strategies 40
3.1.3 Parental Selection 40
3.1.4 Reproduction and Inheritance 41
3.2 Choice of Representation 42
3.3 Characteristics of Fitness Landscapes 42
4 The Challenge of Expansion 44
4.1 Representation and Morphogenesis 44
4.2 Non-random Mating and Speciation 45
4.3 Decentralized, Highly Parallel Models 45
4.4 Self-adapting Systems 45
4.5 Coevolutionary Systems 46
4.6 Inclusion of Lamarckian Properties 46
4.7 Modeling Evolutionary Systems 47
5 Summary and Conclusions 47
3
Evolutionary Computation: Challenges and duties 53 Carlos Cotta and Pablo Moscato
1 Introduction 53
2 Challenge #1: Hard problems for the paradigm – Epistasis and
Parameterized Complexity 55
3 Challenge #2: Systematic design of provably good recombina
tion operators 58
4 Challenge #3: Using Modal Logic and Logic Programming
methods to guide the search 62
4.1 Example 1 63
4.2 Example 2 64
5 Challenge #4: Learning from other metaheuristics and other
open challenges 67
6 Conclusions 69
4
Open Problems in the Spectral Analysis of Evolutionary Dynamics 73 Lee Altenberg
1 Optimal Evolutionary Dynamics for Optimization 76 1.1 Spectral Conditions for Global Attraction 78 1.2 Spectral Conditions for Rapid First Hitting Times 78 1.3 Rapid Mixing and Rapid First Hitting Times 80
1.4 Some Analysis 82
1.5 Transmission Matrices Minimizing 85
1.6 Rapid First Hitting Time and No Free Lunch Theorems 87
2 Spectra for Finite Population Dynamics 87
2.1 Wright-Fisher Model of Finite Populations 88
2.2 Rapid First Hitting Time in a Finite Population 90 3 Karlin’s Spectral Theorem for Genetic Operator Intensity 92
3.1 Karlin’s Theorem illustrated with the Deceptive Trap
Function 93
3.2 Applications for an Extended Karlin Theorem 95
3.3 Extending Karlin’s Theorem 96
3.4 Discussion 98
4 Conclusion 99
5
and Adaptive Memory Metaheuristics
Gary A. Kochenberger, Fred Glover, Bahram Alidaee and Cesar Rego
Solving Combinatorial Optimization Problems via Reformulation 103
1 Introduction 104
2 Transformations 105
3 Examples 106
4 Solution Approaches 108
4.1 Tabu Search Overview 108
5 Computational Experience 109
6 Summary 110
6
Problems in Optimization 115
William G. Macready
1 Introduction 115
2 Foundations 116
3 Connections 120
4 Applications 125
5 Conclusions 127
7
EC Theory - “In Theory” 129
Christopher R. Stephens and Riccardo Poli 8
Asymptotic Convergence of Scaled Genetic Algorithms 157 Lothar M. Schmitt
1 Notation and Preliminaries 162
1.1 Scalars and vectors 162
1.2 Matrices and operator norms 163
1.3 Stochastic matrices 164
1.4 Creatures and populations 167
2 The Genetic Operators 168
2.1 Multiple-spot mutation 169
2.2 Single-cutpoint regular crossover 171 2.3 The fitness function and selection 174 3 Convergence of Scaled Genetic Algorithms to Global Optima 177 3.1 The drive towards uniform populations 177
3.2 Weak ergodicity 179
3.3 Strong ergodicity 180
3.4 Convergence to global optima. 182 3.5 The Vose-Liepins version of mutation-crossover 186
4 Future Extensions of the Theory 187
4.1 Towards finite-length analysis on finite-state machines 187 4.2 Estimates for finite-length genetic algorithms à la Catoni 188
4.3 Adding sampling noise 189
4.4 Further analogy with simulated annealing: parallelism
and sparse mutation 189
4.5 Analysis from inside-out and outside-in 190 4.6 Non-monotone and self-adapting annealing sequences 191 4.7 Discrete vs. continuous alphabets 192 5 Appendix — Proof of some basic or technical results 192 9
of Genetic and Evolutionary Computation
John R. Koza, Matthew J. Streeter and Martin A. Keane
The Challenge of Producing Human-Competitive Results by Means 201
1 Turing’s Prediction Concerning Genetic and Evolutionary Com
putation 202
2 Definition of Human-Competitiveness 202
3 Desirable Attributes of the Pursuit of Human-Competitiveness 203
3.1 Utility 203
3.2 Objectivity 204
3.3 Complexity 204
3.4 Interminability 206
4 Human-Competitiveness as a Compass for Theoretical Work 206 5 Research Areas Supportive of Human-Competitive Results 207 6 Promising Application Areas for Genetic and Evolutionary Com
putation 207
7 Acknowledgements 208
10
Case Based Reasoning 211
Vivek Balaraman
1 Introduction 211
2 Case-Based Reasoning 213
3 Case Memory as an Evolutionary System 216
3.1 A Simple Model of ECM 217
3.1.1 Case-Base 217
3.1.2 Environment 217
3.1.3 Generate Solution 218
3.1.4 Evaluate 219
3.2 Reorganize 219
3.3 Discussion 219
4 Hybrid Systems 224
4.1 Type A - CBR as a memory, EA as the optimizer 225 4.2 Type B - EA as CBR System Parameter Optimizers 226
4.3 Discussion 227
5 Evolving Higher Levels 229
5.1 Schemas 229
5.2 A brief aside on levels of higher expertise 231
5.3 Towards memory based reasoning 232
5.3.1 C-Schemas as Building Blocks 233
6 Conclusions 237
11
The Challenge Of Complexity 243
Wolfgang Banzhaf and Julian Miller
1 GP Basics and State of the Art 245
2 The Situation in Biology 248
3 Nature’s way to deal with complexity 249
4 What we can learn from Nature? 254
5 A possible scenario: Transfer into Genetic Programming 256
6 Conclusion 258
Author Index 261
Index 267
4.1 The Deceptive Trap fitness landscape for three loci with
two alleles. 94
4.2 There is only one attractor at each value but an ‘error
catastrophe’ is evident for 94
4.3 The mean fitness of the population at the global attractor as a function of mutation rate. It decreases in accord
with Karlin’s theorem. 95
10.1 CBR problem solving process 214
10.2 Simple model of evolutionary case memory at genera
tion 218
10.3 ECM as optimizers 223
10.4 Type A: EA Using CBR 225
10.5 TypeB: CBR Using EA 226
10.6 Experiences lead to schema which in turn index new ex
periences 232
11.1 The variation selection loop of GP and other artificial
evolutionary systems. 246
11.2 The primary operations of GP, mutation and crossover, as applied to programs represented by sequences of in
structions. The instructions are coded as integer numbers. 247
11.3 Single cell and multi-cellular system. The environment of a genome is primarily the cell in which it is residing.
Control is exerted both by the cell and its environment via substances (black dots) diffusing around in intra- and extracellular space. The genome in turn tries to influ
ence its environment by providing orders to produce cer
tain substances. If a multi-cellular being is constructed a division and differentiation process is set into motion which leads to a number of cells with a boundary to the outside environment. The organism is the primary en
vironment of a cell, with intra- and extra- organismal
message transfer via molecules (black dots). 250 11.4 Transcription and translation as two important steps in
the process of mapping information from genotype to
phenotype. 252
11.5 The network of data flow on registers as one example of program phenotype. The corresponding program is listed in the text as a linear sequence of instructions.
Adopted from (Brameier, 2003) 257
1.1 Major transitions in evolution; Maynard (Smith and Sza
thmary, 1995) 4
9.1 Eight criteria for saying that an automatically created re
sult is human-competitive 203
This book is a collection of essays, authored by eminent scholars in evo
lutionary computation (EC), artificial intelligence (AI), operations research, complexity theory and mathematics. Each essay revolves around important, interesting and unresolved questions in the field of evolutionary computation.
The book is designed to be a resource to at least three categories of readers.
First of all, graduate students will find this book a rich source of open research issues. Imagine participating in an EC research seminar conducted by some of the best scholars in and around the field! The book also gives experts a chance to compare and contrast their understanding of the fundamental issues in EC with the perspectives of their peers. Finally, to the interested scholar it offers a sample of the kind of problems that are considered worth solving in EC.
Much has been written about how great solutions often have a certain aes
thetic appeal (symmetry, simplicity, originality, unity and so on). In sharp contrast, characteristics of great problems remain something of a mystery. It is useful to think of a problem as existing in at least one of four states: undiscov
ered, unsolved, solved and hibernating. However, truly interesting problems
— great problems — manage a simultaneous, contrary existence in all four quadrants. A great problem, to echo Walt Whitman, is often large and con
tains multitudes. Every mature field has its great problems. Even fields with a progressive tradition, like Physics and Mathematics, have problems that refuse to stay solved. The problem of explaining the directionality of the thermody
namic arrow of time, and the debate over whether mathematical objects are invented or discovered are but two examples that comes to mind. Great prob
lems act as co-ordinate systems for the geography of our imaginations and explain why we do what we do.
So it is gratifying (rather than alarming) that EC is also evolving its own collection of really hard problems. For example, is an evolutionary process an algorithmic process (in the sense of Church-Turing)? Are building blocks theoretical rather than empirical constructs? Which results in EC are dependent on problem representation and which ones independent of it? What precise role does crossover play? Is there a way to unify the different formalisms used
to model evolutionary processes? What are the characteristics of problems solvable by EC? Some of these problems are discussed at length in this volume.
This book grew out of a proposed session for the 2001 International Con
ference on Artificial Intelligence in Las Vegas, Nevada. I had thought that a collection of authoritative essays, each devoted to the description of a sub
stantially unsolved problem in EC, could help bring coherence to the field, clarify its important issues, and provoke imaginations. The session was jok
ingly dubbed the ‘Hilbert session’ in memory of David Hilbert’s outstanding example almost a century ago. Unfortunately, time constraints prevented the session from from going forward. But the highly positive response from the invitees, as well as from others who had heard about the idea, suggested that a book could be an alternate and appropriate forum for implementing the idea.
The stray mentions of Hilbert in some of the essays thus hark back to the ori
gins of this book. Needless to say, the essays were not written with the aim of being either as definitive or as predictive as Hilbert’s talk turned out to be.
The authors in this collection are wonderfully varied in their backgrounds, writing styles and interests. But their essays are related by several common goals: extensions to EC theories, discussion of various formalisms, summaries of the state of the art, and careful speculation on what could be done to re
solve various issues. The essays also leave no doubt that the ferment caused by active trading is producing a watershed event in the marketplace of ideas.
Witness for example, the import of ideas from evolutionary theory into Algo
rithmics (such as: population thinking, inheritance and recombination), and the export of ideas from mathematics and computer science into evolutionary theory (such as: stochastic models, complexity theory, computability). Ide
ally, I would have liked to triple the size of the book, include at least a dozen more authors, and reprint essays from relevant collections. On the other hand, progress is a side-effect of achieving the possible. While the sample of ideas and authors herein is certainly not comprehensive, it is very much representa
tive of what is possible in our field.
EC is a young discipline, and consequently, it is still a field that has the rare chance to be defined in terms of its unsolved problems, rather than its solved ones. No doubt, the many encounters offered in this book, the journeys it will inspire, and the inevitable predilection of problems to get solved, will change this situation in the next few decades. But till then, this book is meant to serve as a beckoning toward the roads still not taken.
Bahrain Alidaee is an associate professor of Operations Management at the Business School, the University of Mississippi. His research interests include combinatorial optimization, heuristic programming, and game theory. He has published more than 40 articles in journals such as Management Science, Trans
portation science, IEEE Transactions, European Journal of OR, Journal of Operational Research, Computers and Operations Research, Production and Operations Management and other journals. He is a member of INFORMS, DSI, APICS, and POMS.
Lee Altenberg is Associate Professor of Information and Computer Sciences at the University of Hawaii at Manoa. His interest is in systems phenomena, and he has focused on the dynamics of evolutionary processes. Of particular interest is the emergence of the representation problem in evolution, the evolu
tion of the genotype-phenotype map, and evolutionary dynamics of modular
ity. His publications bridge the fields of mathematical population genetics and evolutionary computation. Recent civic projects include restoration of dryland Hawaiian biodiversity, reduction of light pollution, and control of alien ungu
lates on Maui.
Vivek Balaraman works as a research scientist in the Artificial Intelligence Group of the Tata Research Development and Design Centre, Pune, India where he has been since 1989. Prior to that he worked at the Knowledge Based Computer Systems Laboratory, Department of Computer Science, Indian Insti
tute of Technology, Madras. Since 1995 he has led the Case-Based Reasoning research and development team at TRDDC. The research has led to the domain independent CBR kernel engine which has been applied successfully on a vari
ety of problems, among them diagnosis, experiential knowledge management and cognitive structured search in tasks like directory assistance and job search at portals. Patent applications have been filed for several aspects of this work.
Vivek’s research interests include machine learning, evolutionary theory and cognitive memory models.
Wolfgang Banzhaf is Associate Professor of Applied Computer Science at the University of Dortmund, Germany. He is lead author of the textbook Genetic Programming — An Introduction and editor-in-chief of the Kluwer journal Ge
netic Programming and Evolvable Machines. He has published more than 80 refereed conference and journal articles.
Carlos Cotta received the M.Sc. and Ph.D. degrees in 1994 and 1998, respec
tively, in Computer Science from the University of Málaga (UMA), Spain. He is currently affiliated to the Department of “Lenguajes y Ciencias de la Com
putación” of the UMA, where he holds an Associate Professorship in Program
ming Languages and Computer Systems. He has been previously appointed as Lecturer (1995–1999), and Assistant Professor (1999–2001) in this institution.
His research interests are primarily in evolutionary algorithms, both from the algorithmic (design techniques, theoretical foundations, parallelism, and hybridization) and the applied (combinatorial optimization, data mining, and bioinformatics) standpoint. He is the author or co-author of over 40 articles on these topics.
He is a member of the European Network of Excellence on Evolution
ary Computing (EvoNet), the European Chapter on Metaheuristics (EU/ME), and the ACM Special Interest Group on Applied Computing (ACM SIGAPP) among other research societies and organizations. He has also served in the Programme Committee of the major conferences in the field (GECCO, CEC, and PPSN among others), and has refereed articles for scientific journals such as the Journal of Heuristics, and IEEE Transactions on Evolutionary Compu
tation among others.
Kenneth A. De Jong is Professor of Computer Science at George Mason Uni
versity, and a member of the research faculty at the Krasnow Institute. He received his PhD at the University of Michigan under the direction of John Holland. Dr. De Jong’s research interests include evolutionary computation, adaptive systems, and machine learning. He is an active member of the evolu
tionary computation research community with a large number of papers, Ph.D.
students, and presentations in this area. He is also involved in the organization of many of the workshops and conferences on evolutionary computation, and the founding Editor-in-chief of the journal Evolutionary Computation, pub
lished by MIT Press. He is currently serving on the executive council of the International Society for Genetic and Evolutionary Computation. Dr. De Jong is head of the Evolutionary Computation Laboratory at GMU, consisting of a group faculty members and graduate students working on a variety of research projects involving the application of evolutionary algorithms to difficult com
putational problems such as visual scene analysis and programming complex
robot behaviors. This group is also involved in extending current evolution
ary computation models to include more complex mechanisms such as speci
ation, co-evolution,and spatial extent. These ideas are being developed to im
prove both the applicability and scalability of current evolutionary algorithms to more complex problem domains. Funding for the lab comes from a vari
ety of sources including DARPA, ONR, NRL, NSF, and local area companies.
Further details can are available at www.cs.gmu.edu/ eclab.
Fred Glover is the MediaOne Chaired Professor in Systems Science at the University of Colorado, Boulder, and Distinguished Researcher and co-founder of the Hearin Center for Enterprise Science at the University of Mississippi. He has authored or co-authored more than three hundred published articles and six books in the fields of mathematical optimization, computer science and artifi
cial intelligence, with particular emphasis on practical applications in industry and government. In addition to holding editorial and advisory posts for jour
nals in the U.S. and abroad, Dr. Glover has been featured as a National Visiting Lecturer by the Institute of Management Science and the Operations Research Society of America and has served as a host and lecturer in the U.S. National Academy of Sciences Program of Scientific Exchange. Professor Glover is the recipient of the distinguished von Neumann Theory Prize, a member of the National Academy of Engineering, and has received numerous other awards and honorary fellowships, including those from the American Association for the Advancement of Science (AAAS), the NATO Division of Scientific Affairs, the Institute of Operations Resarch and Management Science (INFORMS), the Decision Sciences Institute (DSI), the U.S. Defense Communications Agency (DCA), the Energy Research Institute (ERI), the American Assembly of Colle
giate Schools of Business (AACSB), Alpha Iota Delta, and the Miller Institute for Basic Research in Science. He serves on the advisory boards of several organizations and is co-founder of OptTek Systems, Inc.
Martin A. Keane received a Ph.D. in Mathematics from Northwestern Uni
versity in 1969. He worked for Applied Devices Corporation until 1972, in the Mathematics Department at General Motors Laboratory until 1976, and was Vice-President for Engineering of Bally Manufacturing Corporation until 1986. He is currently chief scientist of Econometrics Inc. of Chicago and a consultant to various computer-related and gaming-related companies.
Gary A. Kochenberger is the Professor of Operations Management, Univer
sity of Colorado at Denver (since 1989). In recent years, his focus has been on problems of a combinatorial nature as commonly found in logistical man
agement, operations management, and related areas. He has published three
books and more than 40 refereed articles in top journals in his field includ
ing Management Science, Mathematical Programming, Journal of Optimiza
tion Theory and Applications, Operation Research, Computers and Operations research, Naval Research Logistics Quarterly, Journal of the Operational Re
search Society, Interfaces, Operations Research Letters, and the Journal of the Production and Operations Management Society and Transportation Science.
Moreover, he is actively engaged in several major journals including positions as area editors for both INTERFACES and the Journal of the Production and Operations Management Society (POMS).
John R. Koza received his Ph.D. in Computer Science from the University of Michigan in 1972 under the supervision of John Holland. He was co-founder, Chairman, and CEO of Scientific Games Inc. from 1973 through 1987 where he co-invented the rub-off instant lottery ticket used by state lotteries. He has taught a course on genetic algorithms and genetic programming at Stanford University since 1988. He is currently a consulting professor in the Biomedical Informatics Program in the Department of Medicine at Stanford University and a consulting professor in the Department of Electrical Engineering at Stanford University.
William Macready is a senior scientist with the Research Institute for Ad
vanced Computer Science at NASA Ames Research Center. William has pub
lished on topics including optimization, landscapes, molecular evolution, ma
chine learning, economics, and methods of quantifying complexity. He is in
terested in both the theoretical and practical aspects of optimization. Before joining RIACS/NASA, William worked in industry solving optimization prob
lems in logistics, supply chains, scheduling, and designed efficient optimiza
tion approaches for clearing high-dimensional marketplaces.
Anil Menon received his Ph.D in Computer Science from Syracuse University in 1998. His thesis Replicators, Majorization and Probabilistic Databases:
New Approaches For The Analysis Of Evolutionary Algorithms was awarded the Syracuse All-University Best Doctoral Thesis Award in 1999. His research interests lie in the areas of evolutionary computation and nonlinear optimiza
tion, he has been published in the peer-reviewed International J. of Neural Networks, IEEE Transactions On Evolutionary Computation, and the Foun
dations of Genetic Algorithms. He has more than eight years of software de
velopment experience in a wide variety of industries including, advanced data access applications, supply chain solutions, and computer aided design and manufacturing, His technical expertise lie in the areas of distributed databases, process optimization, and automated code generation. Till recently, he was a
Distinguished Engineer at Whizbang Research Labs, based in Provo, Utah. He currently consults for ProductSoft, a Pittsburgh based software startup.
Julian Miller is a lecturer in the School of Computer Science, University of Birmingham. He is author of around 75 research publications. He has an in
ternational reputation in the research fields of evolvable hardware and genetic programming. He is a regular programme chair and session chair for inter
national conferences on evolvable hardware, and Genetic Programming, and evolutionary computation. He is an associate editor of the Journal of Genetic Programming and Evolvable Machines and IEEE Transactions on Evolution
ary Computation. He teaches advanced MSc modules on Quantum and Molec
ular Computation and a Nature Inspired Design and a first-year undergraduate course on Artificial Intelligence Programming.
Heinz Mühlenbein is research manager of the team ‘Adaptive Sytems’ at the Institut of Autonomous intelligent Systems of the Fraunhofer Gesellschaft in Germany. He has worked in the areas of time-sharing operating systems, com
puter networks, parallel computing, and since 1987, soft computing. He is on the editorial board of journals in physics, parallel computing, heuristics, and evolutionary computation. In addition he has published more than 30 refereed journal articles.
Pablo Moscato is currently affiliated with The University of Newcastle, Aus
tralia. A native from Argentina, he received a degree in Physics from Uni
versidad Nacional de La Plata in 1987 and a PhD degree from UNICAMP, Brazil. In 1988-89 was a Visiting Graduate Student at the California Insti
tute of Technology and a member of the core research team of the Caltech Concurrent Computation Program where he first introduced the denomination of “memetic algorithms” in the computing literature. He has been Visiting Professor at Universidad del Centro de la Provincia de Buenos Aires, Tandil, Argentina (1995-1996), and UNICAMP, Campinas, Brazil (1996) where he lectured on metaheuristics for combinatorial optimization. He has published in Journal of Computer and System Sciences, Physics Letters A, Lecture Notes in Computer Science, Annals of Operations Research, Applied Mathematics Letters, Neural Networks World, Chaos, Solitons and Fractals, Production Planning & Control, INFORMS Journal on Computing, Pesquisa Operacional, and European Journal of Operations Research as well as several international conferences. He is a member of the Editorial Board of The Journal of Mathe
matical Modelling and Algorithms and a member of the Program Committee of several international conferences. He is author or co-author of twelve chap
ters of books. He also acted as co-ordinating editor of the section dedicated
to Memetic Algorithms of “New Ideas in Optimization”, McGraw-Hill, UK, 1999.
Riccardo Poli is a Professor of Computer Scince at the University of Essex.
He has co-authored the book Foundations of Genetic Programming and around 130 refereed publications (including 10 conference proceedings) on genetic programming, genetic algorithms, image processing and neural networks and other areas. He is an associate editor of Evolutionary Computation and of the Journal of Genetic Programming and Evolvable Machines. He has been pro
gramme committee member of over 40 international events and has presented invited tutorials at 8 international conferences.
César Rego received his Ph.D. in Computer Science from the University of Versailles and INRIA - France, after earning a MSc in Operations Research and Systems Engineering from the Technical School (IST) of the University of Lis
bon. His undergraduate degree in Computer Science and Applied Mathematics is from the Portucalense University in Portugal. Part of his academic career was done in the Portucalense University and he also taught at IST and Faculty of Sciences of the University of Lisbon. Dr. Rego received an award from the Portuguese Operational Research Society (APDIO) for his MSc thesis. He also received the IFORS-Lisbon award for the best international paper published by members of APDIO, an investigation over a four-year period. Finally, he received an award as Researcher/Scholar of the year in School of Business, MIS/POM area, University of Mississippi. Professor Rego’s publications have appeared in books on metaheuristics and in leading journals on optimization such as European Journal of Operational Research (EJOR), Journal of Oper
ational Research Society (JORS), Parallel Computing, and Management Sci
ence. He has developed some of the most efficient algorithms that currently exist for the Traveling Salesman and Vehicle Routing Problems. In the prac
tical realm, he has designed and implemented computer software for solving real-world problems for several major companies. His main research interest is the creation and empirical validation of optimization algorithms for solving complex and practical problems. He is a member the APDIO, the INFORMS, a senior researcher of the Hearin Center for Enterprise Science (HCES) and As
sociate Professor of Management Information Systems and Operations Man
agement in the School of Business of The University of Mississippi.
Lothar M. Schmitt teaches Mathematics and Computer Science at The Uni
versity of Aizu (Japan). He holds the Dr.rer.nat. title from the Universitaet des Saarlandes (Saarbruecken) and the Dr.rer.nat.habil. title from Universitaet Osnabrueck and is currently associate professor in Aizu. His work includes
contributions in functional analysis, operator algebra theory, non-commutative integration, quantum physics, biomechanical modeling, genetic algorithms and optimization, language analysis and UNIX-based interactive teaching systems.
Otherwise, he enjoys family life, swimming, playing the piano, the arts and fine dining. In 2003, he is listed in “Who’s Who in the World.”
Chris Stephens is Professor at the Institute for Nuclear Sciences of the UNAM (Universidad Nacional Autonoma de Mexico) - the oldest university in the Americas. He has had visiting positions at various leading academic insti
tutions, including the Weizmann Institute, the Joint Institute for Nuclear Re
search, Dubna, the University of Birmingham and the University of Essex. He is also a founding partner of Adaptive Technologies Inc. and Adaptive Tech
nologies SA de CV - research companies dedicated to the production of agent- based technologies for dynamical optimization in finance and industry. Chris’s research interests are very broad, having published over 70 research articles in a wide array of international journals - ranging from Classical and Quantum Gravity to the Journal of Molecular Evolution.
Matthew J. Streeter received a Masters degree in Computer Science from Worcester Polytechnic Institute in 2001. His Masters thesis applied genetic programming to the automated discovery of numerical approximation formu
lae for functions and surfaces. His primary research interest is applying genetic programming to problems of real-world scientific or practical importance. He is currently working at Genetic Programming Inc. as a systems programmer and researcher.
TOWARDS A THEORY OF ORGANISMS AND EVOLVING AUTOMATA
Open Problems and Ways to Explore
Heinz Mühlenbein
FhG-AiS D-53754 Sankt Augustin [email protected]
Abstract We present 14 challenging problems of evolutionary computation, most of them derived from unfinished research work of outstanding scientists such as Charles Darwin, John von Neumann, Anatol Rapaport, Claude Shannon, and Alan Tur
ing. The problems have one common theme: Can we develop a unifying theory or computational model of organisms (natural and artificial) which combines the properties structure, function, development, and evolution? There exist theories for each property separately as well as for some combinations of two. But the combination of all four properties seems necessary for understanding living or
ganisms or evolving automata. We discuss promising approaches which aim in this research direction. We propose stochastic methods as a foundation for a unifying theory.
1. INTRODUCTION
The aim of this book is very ambitious. Its title is not: important problems of evolutionary computation, but Hilbert problems in evolutionary computation.
What makes Hilbert’s problems so famous and unique? Hilbert designed his problems with the goal that “they could serve as examples for the kinds of problems the solutions of which would lead to advancements of disciplines in mathematics.” If we have a closer look at Hilbert’s twenty-three problems today, then we observe that some of the problems indeed lead to important research, but a few of them did not. One of the reasons seems to be how the problems have been formulated. Most of them are well defined, but some are more vaguely posed, making a solution difficult.
In fact, the paper became famous because of question number two: Can it be proven that the axioms of arithmetic are consistent? Hilbert’s question is a
sub-problem of the general research program Hilbert had in mind: Can math
ematics be axiomatized? The general problem was taken on by Russel and Whitehead and lead to three volumes of the Principia Mathematica. Gödel dealt with the more specific problem two and proved that the answer is nega
tive. This put an end to the effort of Russel and Whitehead. The implication of Gödel’s result with regard to mathematics and the theory of computation in general is still a subject of hot discussions.
In contrast, problem number six just reads: Can physics be axiomatized? In the explanation of the question Hilbert writes: “to axiomatize those physical disciplines, in which mathematics already plays a dominant role; these are first and foremost probability and mechanics.” To our surprise we see the calculus of probability as a part of physics! A closer inspection reveals that Hilbert’s moderate goal was a mathematically sound application of probability to kinetic gas theory. This research has been carried out by physicists, but without ever referring to a Hilbert problem. It lead to statistical physics as it appears today.
My goal is modest. I will propose problems, mainly in evolutionary com
putation, and name each after a famous scientist who has formulated or inves
tigated the problem. This does not imply that the problem so named is the most important the scientist has worked on. Nor do I claim that the scientist has considered the problem to be the most important one he has worked on.
I only want to demonstrate that most of the challenging problems have been identified very early and are with us for quite a time. And my second message is: we have to look much more often into older papers. Older scientific pa
pers should not be considered as “fossils”. It is a fundamental misconception that science is continuously accumulating all the important available knowl
edge and condensing the knowledge in surveys or textbooks. Many important scientific ideas and papers enter main stream science after 20 or more years.
I will consider in the paper both – natural and artificial organisms. The emphasis will be on artificial automata. In order not just to summarize the problems, I will describe in the more technical sections 11 till 13 a theory I consider as a promising candidate for solving some of the problems presented.
It is the theory of probability, used and extended in scientific disciplines as dif
ferent as probabilistic logic, statistical physics, stochastic dynamical systems and function optimization using search distributions. These sections will be fairly selfish, because in selecting from the huge available literature the work of my research group will be over-represented.
2. EVOLUTIONARY COMPUTATION AND THEORIES OF EVOLUTION
The goal of evolutionary computation is to make the development of pow
erful problem solving programs easier. There have been tried at least three approaches to achieve this goal.
1 Use a theory - develop a theory of problem solving and implement it on a computer
2 Copy the brain - analyze the human brain and make a copy of it on a computer
3 Copy natural evolution - analyze natural evolution and implement the most important evolutionary forces on a computer
In the history of artificial intelligence research one of the three approaches was dominant at any one time. Evolutionary computation belongs to the third approach. Today this approach is gaining momentum. It relies on theories of evolution and of computation. The theory of computation is well advanced, so the problems of evolutionary computation lie in theories of evolution. If there existed a convincing constructive theory of evolution, then evolutionary computation would be just a matter of implementation - which of the major evolutionary forces to implement in what detail.
But do we possess a constructive theory of evolution? Here the opinions dif
fer extremely. The main stream theory of evolution is called New or Modern Synthesis. Its followers claim that it reconciles Darwin’s idea of continuous small variations with the concept of gene flows derived from population genet
ics. The second major force of the Modern Synthesis is still Darwin’s concept of natural selection. But are these two forces sufficient to explain the wonders of evolution at least in some broad terms?
There is no doubt that the modern synthesis is able to explain the change of gene frequencies on a small time scale. If there is enough diversification, then the theory correctly predicts further changes for a short time. But can it explain evolution for a long time? Here the crucial question is: How could it come to such a diversification, starting from a tiny cell? I like to formulate the problem with Darwin’s famous ending sentence of The Origin of Species by Means of Natural Selection ((Darwin, 1859)).
“There is grandeur in this view of life, with its several powers, having been originally breathed into a few forms or into one; and that, whilst this planet has gone cycling on according to the fixed laws of gravity, from so simple a beginning endless forms most beautiful and most wonderful have been, and are being, evolved.”
Let me be more specific and cite some major problems which a theory of evolution would have to explain. Maynard (Smith and Szathmary, 1995), have called them the the major transitions in evolution (see table 1.1).
The authors “solve” some of the problems with a very narrow version of the modern synthesis. “We are supporters of the gene centered approach pro
posed by Williams and refined by (Dawkins, 1989).” In the gene centered approach, also called the selfish gene concept, the genes are the major actors.
They possess an internal force to proliferate as much as possible.
This caricature of a theory of evolution is used by the authors to explain the transition from solitary individuals to colonies, for example. The argument is as follows: If a female produces two offspring, but females can produce 3n offspring, then cooperation between the females pays off. Even if there is a fight between females and one becomes a queen, cooperation is still preferred
of is larger than 2). Thus in the gene centered analysis a colony with a single queen has a selective advantage.
There are many flaws in the selfish gene concept. It is not constructive, it does not investigate if the selection advantage of a particular gene can be re
alized in a phenotype. Rabbits with wings would obviously have a selective advantage. Why did it not happen? Two genes can also oppose each other - gene 1 might increase by action and gene 2 by the opposite action Which gene wins? Consider a female and its offspring as an example. The off
spring are threatened. Should the mother protect the offspring, even on the risk of her life? The notorious formula of Hamilton gives the result that the mother should sacrifice her life if more than two offspring are threatened (Maynard (Smith and Szathmary, 1995)). Hamilton argues as follows: in each offspring there are only one half of the genes of the mother. Thus the genes of the mother multiply if she protects at least three offspring. Ironically Darwin itself has de
voted a whole chapter of his “The Origin of Species” to the problem insect colonies pose to natural selection. His explanation is constructive. He shows how many small changes in behavior can lead to very peculiar behavior, even to slave making ants! This example shows dramatically the extreme simplifi
cation done by the selfish gene concept. It is my strong opinion that the selfish gene concept does not enrich Darwin’s theory, but reduces it to a caricature.
The selfish gene concept has been opposed by a small group in biology, most notably by the late Stephen J. Gould. Recently even philosophers of science formulate a basic critic. I just cite (Griffiths, 2002). “The synthetic theory bypassed what were at the time intractable questions of the actual re
lationship between stretches of chromosomes and phenotypic traits. Although it was accepted that genes must, in reality, generate phenotypic differences through interaction with other genes and other factors in development, genes were treated as black boxes that could be relied on to produce phenotypic vari
ation with which they were known to correlate.”
I will discuss this problem later with my proposal of a system theory of evolution. The major conclusion of this section is: there exists no general the
ory of evolution today. The “theory” its proponents call “Modern Synthesis”
is an extremely simplified version of Darwin’s theory. It separates organisms and environment. Natural selection is modeled by a fitness function, whereas Darwin used the term only in a metaphoric sense. In fact, Darwin noticed the misinterpretation of his theory even during his life. He wrote in the last (1872) edition of “The Origin of Species”: “As my conclusions have lately been much misrepresented, and it has been stated that I attribute the modifica
tion of species exclusively to natural selection, I may be permitted to remark that in the first edition of this work, and subsequently, I placed in a most con
spicuous position — namely at the close of the Introduction — the following words: “I am convinced that natural selection has been the main but not the ex
clusive means of modification.” This has been of no avail. Great is the power of steady misinterpretation.”
Therefore evolutionary computation has to be largely experimental. This was already pointed out by John (von Neumann, 1954). “Natural organism are, as a rule, much more complicated and subtle, and therefore much less well un
derstood in detail, than are artificial automata. Nevertheless, some regularities, which we observe in the organization of the former may be quite instructive in our thinking and planning of the latter; and conversely, a good deal of our experiences and difficulties with our artificial automata can be to some extend projected on our interpretations of natural organisms.”
3. DARWIN’S CONTINENTAL CYCLE CONJECTURE
I will describe my first problem in Darwin’s terms. In the chapter “Cir
cumstances favorable to Natural Selection” Darwin writes: “A large number of individuals by giving a better chance for the appearance within any given period of profitable variations, will compensate for a lesser amount of vari
ability in each individual, and is, I believe, an extremely important element of success.”
On the other hand Darwin observes that a large number of individuals in a large continental area will hinder the appearance of new adaptations. This happens more likely in small isolated areas. He writes: “Isolation, also, is an important element in the process of natural selection. In a confined or isolated area, if not large, the organic and inorganic conditions of life will be in a great degree uniform; so that natural selection will tend to modify all individuals of a varying species throughout the area in the same manner in relation to the same conditions. But isolation probably acts more efficiently in checking the immigration of better adapted organisms. Lastly, isolation, by checking immigration and consequently competition, will give time for any new variety to be slowly improved.”
Darwin then continues: “Hence an oceanic island at first sight seems to have been highly favorable for the production of new species.” But Darwin notes a conflict: “to ascertain whether a small isolated area or a large open area like a continent, has been most favorable for the production of new organic forms, we ought to make the comparison within equal times; and this we are incapable of doing. ”
Despite of the above observation Darwin concludes: “I conclude, that for terrestrial productions a large continental area, which will probably undergo many oscillations of level, and which consequently will exist for long periods in a broken condition, will be the most favorable for the production of many new forms of life, likely to endure long and spread widely.” Darwin reasons as follows: “For the area will first have existed as a continent, and the inhabitants, at this period numerous in individuals and kinds, will have been subjected to very severe competition. When converted by subsidence into large separate islands, there will still exist many individuals of the same species on each island;. . . and time will be allowed for the varieties in each to become well modified and perfected. When by renewed elevation, the islands shall be re
converted into a continental area, there will be again severe competition: the most favored or improved varieties will be enabled to spread: there will be much extinction of the less improved forms . . .”
I am very impressed about Darwin’s continental cycle conjecture, which he made much earlier than Alfred Wegener in geology. Therefore I dedicate my first problem to Darwin.
Problem 1 [Darwin]: Can we demonstrate or even prove the correctness of Darwin’s Continent-Island cycle conjecture ?
The reader should have observed how carefully Darwin discusses the ar
guments. I strongly recommend to read Darwin’s “The Origin of Species”.
The most profound critique of modern “Darwinism” can be found in Darwin’s book!1
It seems difficult to test Darwin’s conjecture in nature. I propose therefore to use simulations as a first step. I have used the iterated prisoners dilemma game to investigate problem 1 ((Mühlenbein, 1991a)). The results indicate that Darwin’s conjecture might be correct. But the simulation model needs a lot more refinement.
Darwin mentions at many places of the “Origin” that space is as important for evolution as time. This has been shown in the context of genetic algorithms by (Mühlenbein, 1991b). Space is also an important element of the shifting balance theory of evolution proposed by (Wright, 1937). Without referring to Darwin a subset of the problem, that is the difference of the evolution in a large continent and small isolated islands, has been recently investigated by (Parisi and Ugolini, 2002).
4. THE SYSTEM VIEW OF EVOLUTION
The next set of problems I will derive more abstract. The major weakness of “Darwinism” in the form of the modern synthesis is the separation of the individuals and the environment. In this model each individual (mainly characterized by its genes) is assigned a fitness predicting the performance of this individual within the environment E and given the other individuals.
This can be written as:
It seems impossible to obtain numerical values for the fitness. Therefore theoretical biology has made many simplifications. The environment is kept fixed, i.e the influence of other individuals is described by some averages of the population, etc.. The shortcomings of the dichotomy individual-environment in the Modern Synthesis have already been discussed.
The problem is still more difficult because each individual is in addition devel
oping in a close interaction with its environment. The development problem has been addressed recently by (Oyama, 2000), in her developmental system theory. Unfortunately the theory is very informal, it has been formulated from a philosopher’s point of view. Therefore I will describe the next problem as it has been stated in the final address of Anatol Rapaport, the then retiring presi
dent of General System Science Society ((Rapaport, 1970)).
1In addition I recommend the essays of Stephen J. Gould.
Problem 2 [Rapaport+1]: Can we formulate a theory of organisms, which incorporates being, acting, evolving, and developing?
I have named the problem Rapaport+1 because Rapaport identified only three properties. He combined evolving and developing into a single property becoming. The problem needs an explanation. It goes back to (Whitehead, 1948). In his book “Science and the Modern World” Whitehead warned that the store of fundamental ideas on which the then contemporary science was based was becoming depleted. Whitehead suggested that the concept of or
ganism, hitherto neglected in physical science, might be a source of new ideas.
Whitehead tried to define what an organism characterizes.
We will describe the definition of Rapaport. “According to a soft definition, a system is a portion of the world that is perceived as a unit and that is able to maintain its identity in spite of changes going on in it. An example of a system par excellence is a living organism. But a city, a nation, a business firm, a university are organisms of a sort. These systems are too complex to be described in terms of succession of states or by mathematical methods.
Nevertheless they can be subjected to methodological investigations.”
Rapaport then defines: “Three fundamental properties of an organism ap
pear in all organism-like systems. Each has a structure. That is, it consists of inter-related parts. It maintains a short-term steady state. That is to say, it reacts to changes in the environment in whatever way is required to maintain its integrity. It functions. It undergoes slow, long term changes. It grows, develops, or evolves. Or it degenerates, disintegrates, dies.
Organisms, ecological systems, nations, institutions, all have these three attributes: structure, function, and history, or, if you will, being, acting, and becoming.”
Rapaport’s becoming captures both – the development of an organism from the fertilized egg to the grown-up organism, and the evolution of the species in a succession of many generations. There is no doubt that the relationship between the two properties is a very close one. Ernst Haeckel even postulated in 1890 a biogenetic law: Individual development is a shortened recapitulation of the history of the phylum. Subsequent research has shown that there is some truth in the law, but as a general statement it is incorrect. In my opinion it is very important to distinguish between the development of an individual and the evolution of a species.
To my knowledge, Rapaport’s talk did not lead to a scientific effort to build such a theory of organisms. The reader will guess the reason: it is the sheer complexity of the task! Instead research in biology remained concentrated on a single property or to a combination of two properties. Thus population ge
netics combines being and evolving, population dynamics combines being and
acting. The developmental system theory mentioned earlier combines being and developing.
The investigation of the above problem leads to another problem: In what language should we frame a theory of organisms? Three approaches can be tried:
The descriptive approach, using natural language The micro-simulation approach
The mathematical approach
Today the descriptive approach has gained momentum, characterized by the developmental system theory mentioned above (Oyama, 2000). Artificial Life uses micro-simulation. But in micro-simulations it is very difficult to distin
guish between the microscopic event and the more general pattern happening in many simulations. Rapaport and, earlier, von Neumann advocated the math
ematical approach. I go a step further and propose stochastic system theory as the research foundation. Stochastic analysis has been successfully used in pop
ulation genetics for at least 75 years. But population dynamics is still mainly investigated with the help of deterministic differential equations. Thus I parti
tion Rapaport’s problem into three problems.
Problem 3a: Can we develop a stochastic system theory, combining the properties being and acting of organisms or automata in a 2-d space?
Problem 3b: Can we develop a stochastic system theory, combining the properties being and developing of organisms or automata in a 2-d space?
Problem 3c: Can we develop a stochastic system theory, combining the as
pects being, acting and evolving of organisms or automata in a 2-d space?
The answer to the first question is a definite yes. It is already an active area of research. We will discuss the state of the art in stochastic analysis in the technical sections 11 till 13. Problem 3b was first investigated by von Neumann.
5. VON NEUMANN’S SELF-REPRODUCING AUTOMATA
Von Neumann started his research with the concept of “complification”. He used the term very informally. We will proceed in the same way. It is outside the scope of this paper to discuss all the measures proposed for complexity.
Also the term automaton will be used in a broad manner. Von Neumann ob
served: “If automaton A can produce B, then A in some way must have con
tained a complete description of B. In this sense some decrease in complexity
must be expected as one automaton makes another automaton.” But organisms reproduce themselves with no decrease in complexity. Moreover, organisms are indirectly derived from others which had lower complexity.
Problem 4 [von Neumann]: Can we construct automata which are able to produce automata more complex than themselves?
Von Neumann tried several approaches to enable a scientific investigation of the above problem. The main theory was collected by Burns and expended into a theory of self-reproducing automata ((Burns, 1970)). But it is more in
structive to look at von Neumann’s own description, summarized in the article
“The General and Logical Theory of Automata” ((von Neumann, 1954)). Von Neumann started his research with a result of Turing. Turing wanted to give a precise definition of what is meant by a computing automaton. His solution was the Universal Turing Machine UTM. It consists of an automaton reading and writing symbols on an infinite tape. Von Neumann decided that his au
tomaton should have the power to simulate the UTM in a discrete cellular 2-d space. Thus he investigated the problem how to construct an automaton which reproduces itself in 2-d space and has the power of UTM.
Von Neumann’s construction proceeded as follows:
(a) Construct an automaton A, which when furnished the description of any other automaton in terms of appropriate functions, will construct that entity.
(b) Construct an automaton B, which can make a copy of any instruction that is furnished to it. This facility will be used when furnishes a description of another automaton.
(c) Combine the automata A and B with a control mechanism which does the following. will first cause A to construct the automaton which is de
scribed by Next will cause B to copy the instruction Finally will separate this construction from the system
(d) Form an instruction which describes this automaton D, and insert into A within D. Call the aggregate which now results E.
E is clearly self-reproducing. But E cannot do anything besides reproduc
tion. It needs a program. Therefore von Neumann proposed an extension:
Replace the instruction by an instruction which describes automa
ton D plus another automaton F. This automaton reproduces itself and then behaves like automaton F. Now if a “mutation” within the F part takes place, it changes into This “mutant” is still self-reproductive.
Von Neumann believed that with this construction he had made crude steps in the direction of a systematic theory of automata, especially towards forming a rigorous concept of what constitutes “complication.” At a first glance, the construction seems to be the solution of the automatic programming problem.
But why did von Neumann’s self-reproducing automata not have any practical
relevance? The answer is simple: The construction does not solve the most important problem: How do the programs get into the machine? The develop
ment of programs is the problem, not their self-reproduction. Von Neumann’s automata can in principle compute anything, but the programs have to be pro
vided from the outside! Who provides these descriptions? A single built-in program F is surely not enough, because von Neumann did not introduce se
lection. Therefore the value of the mutant program F' for problem solving is not checked. Thus von Neumann solved only part of the problem. Therefore we extend problem 4.
Problem 5: What conditions are required to enable von Neumann’s au
tomata to grow in complexity without external interventions?
A worthwhile extension of von Neumann’s approach would be to use a pop
ulation of automata which interact with each other and which have to solve a set of problems to survive and produce offspring. Thus I believe that for a solution of problem 5 one needs both, Turing and Darwin. Turing provides the concept of a universal automaton and Darwin provides the concept of a changing environment metaphorically leading to natural selection.
The importance of von Neumann’s construction for today’s research has also been emphasized by (McMullin, 2001).
6. TURING’S INTELLIGENT MACHINE
Von Neumann’s approach of using self-reproduction and the Universal Tur
ing Machine was not the only method proposed to build intelligent machines.
In fact, von Neumann discussed the use of artificial neural networks as another possibility. Before I describe this work, it is instructive to discuss how Tur
ing himself approached the problem in his article “Computing machinery and intelligence” ((Turing, 1950)). At first Turing defined the concept of intelli
gence. A machine is intelligent if it passes a test Turing defined precisely: the Turing test is an “imitation” game, played by three objects A, B and C. C is the interrogator, A or B might be a machine. The machine passes the test if the interrogator is not able to find out that a machine answers to his questions.
This gives our next problem.
Problem 6 [Turing]: Is it possible to build machines which pass the Turing test?
Turing believed that the answer to the above question is positive and pro
posed a method to construct such a machine. It is described in the section
“Learning Machines” of the above cited paper. Turing’s proposal seems to be almost unknown, although it is contained in this well-known article. I find
